Method for measuring the resistance and the inductance of a line

ABSTRACT

The method described here makes it possible to determine the impedance of a line ( 1 ) by measuring the voltage (u) applied across the line and the time derivative of the current (i) flowing through the line. The measured values of the differentiated current are not integrated in this case, but rather are substituted directly, together with the measured voltage values, into an equation system from which the values of the inductance (L) and the resistance (R) of the line ( 1 ) can be estimated. In this way, integration of the values of the differentiated current is obviated.

FIELD OF THE INVENTION

[0001] The invention relates to a method for measuring the resistanceand the inductance of a line according to the preamble of claim 1. Suchmethods are used e.g. for short circuit detection in electricity supplylines.

BACKGROUND OF THE INVENTION

[0002] In order to measure the impedance and, in particular, theresistance and the inductance of a line, the current flowing in it andthe voltage applied across it are generally measured in a time-resolvedfashion and converted into the required values by computation. For themeasurement of currents, especially in the case of electrical powerlines, use has also recently been made of a Rogowski coil, i.e. a coilthat extends around the line carrying the current and measures thederivative of the current with respect to time. In order to find thecurrent, this derivative needs to be integrated with respect to time.This necessitates additional numerical calculation and can lead toinaccuracies (clipping effects, phase offset).

SUMMARY OF THE INVENTION

[0003] It is therefore an object of the invention to provide a method ofthe type mentioned in the introduction, which gives a maximally accurateand simple way of measuring the inductance and the resistance.

[0004] This object is achieved by the method as claimed in claim 1.Instead of integrating the values from the Rogowski coil, the measuredvalues are hence substituted directly into an equation system thatdelivers the desired results.

[0005] Preferably, a sizeable number of measurements are used, so thatthe equation system is over-determined. The resistance and theinductance can then be found by adjustment computation. The method ofleast squares is preferably employed, which can be done by means ofsimple matrix inversion or recursively.

BRIEF DESCRIPTION OF THE DRAWING

[0006] Further configurations, advantages and applications of theinvention are contained in the dependent claims and from the followingdescription with reference to FIG. 1. The figure shows an equivalentcircuit diagram of a line to be measured.

[0007]FIG. 1 shows the equivalent circuit diagram of a line 1 with aresistance R and inductance L. It may, for example, be an electricalpower line that is short-circuited at a point 2.

[0008] In order to measure the voltage u(t) across the line 1, avoltmeter 3 is provided. It may, for example, be an electro-opticalvoltage transformer that is capable of reproducing the value of thevoltage u(t) directly (possibly to within a known calibration constant).In order to measure the current i(t) in the line 1, a currentdifferential meter 4 is used. This comprises a Rogowski coil 5, whichproduces a voltage proportional to the time derivative ∂/∂t of i(t).Accordingly, the current differential meter 4 produces measurements thatcorrespond (to within a known calibration constant) to the value∂i(t)/∂t.

[0009] The meters 3, 4 are operated at a sampling rate ƒ_(A)=1/T_(A) anddeliver a series of voltage values u₀, u₁, . . . and a series of valuesl₀, l₁, . . . l_(N) of the differentiated current, with u_(k)=u(kT_(A))and l_(k)=∂l/∂t|t=kT_(A). The resistance R and the inductance L are tobe determined from these values.

[0010] For the voltage u(t), the following applies: $\begin{matrix}{{u(t)} = {{R \cdot {i(t)}} + {{L \cdot \frac{\partial}{\partial t}}{i(t)}}}} & (1)\end{matrix}$

[0011] Using the Rogowski coil, the time derivative of the current i(t)is measured, i.e. the quantity $\begin{matrix}{{i(t)} = {\frac{\partial}{\partial t}{i(t)}}} & (2)\end{matrix}$

[0012] The current i(t) can be calculated by integrating (2).Substitution into (1) hence leads to:

u(t)=R·∫l(τ)dτ+L·i(t)  (3)

[0013] Equation (3) can be converted into a discretized form.Preferably, the Laplace transform u(p) is formed, and a bilineartransformation p=A·(z−1)/(z+1), where A takes the value A≅2/T_(A) in thecase of the Tustin (bilinear) approximation. A can more accurately bedescribed as $\begin{matrix}{{A = {\omega_{0} \cdot {\cot ( {\frac{\omega_{0}}{2} \cdot \frac{T_{A}}{2}} )}}},} & \text{(3a)}\end{matrix}$

[0014] where ω₀ is the frequency at which the digital approximation isto agree with the analog values. When formula (3a) is used instead ofA=2/T_(A), the approximation by the digital model agrees accurately withthe analog model at the angular frequency ω₀. In practice, w₀=2πƒ_(N) isselected, where ƒ_(N) is the mains frequency of the voltage along theline to be surveyed.

[0015] Discretization of Equation (3) approximately leads to:

u _(n) =u _(n−1)+α₀ ·l _(n)+α₁ ·l _(n−1)  (4)

[0016] where u_(n) and u_(n−1) are two successively sampled voltagevalues and l_(n) and l_(n−1) are two successively sampled values of thedifferentiated current. The parameters α₀ and α₁ are given by$\begin{matrix}{{\alpha_{0} = {\frac{R}{A} + L}}{\alpha_{1} = {\frac{R}{A} - L}}} & (5)\end{matrix}$

[0017] Equation (4) provides a basis for finding the required values Rand L, or the parameters α₀ and α₁. To that end, N measurements of thevoltage {u₁, u₂, . . . u_(N)} and of the differentiated current {l₁, l₂,. . . l_(N)}, with N≧3, are needed in order to set up, from (4), anequation system with N−1 equations for a₀ and a₁, so that the parameterscan be calculated. This procedure has the advantage that directintegration of Equation (3) can be circumvented.

[0018] From the parameters α₀ and α₁, the resistance R and theinductance L can then be determined from $\begin{matrix}{{R = {A \cdot \frac{( {\alpha_{0} + \alpha_{1}} )}{2}}}{L = \frac{\alpha_{0} - \alpha_{1}}{2}}} & (6)\end{matrix}$

[0019] Preferably, N>3, i.e. more than two equations are set up, so thatover-determination of the equation system is obtained. The parametervalues can, in this case, be determined with high accuracy by means ofadjustment computation.

[0020] For example, the parameters α₀ and α₁, can be determined byapplying the method of least squares to the linear equations (4) forn=1. . . N, which in vector notation leads to the following solution:$\begin{matrix}{{\Theta = {\begin{pmatrix}\alpha_{0} \\\alpha_{1}\end{pmatrix} = {\lbrack {\sum\limits_{n = 1}^{N}\quad {m_{n}m_{n}^{T}}} \rbrack^{- 1} \cdot {\sum\limits_{n = 1}^{N}\quad {m_{n}y_{n}}}}}}{with}{m_{n} = \begin{pmatrix}i_{n} \\i_{n - 1}\end{pmatrix}}} & (7)\end{matrix}$

[0021] and y_(n)=(u_(n)−u_(n−1))

[0022] In this, the first multiplicand on the right-hand side ofEquation (7) is the inverse of a 2×2 matrix produced from the valuesl_(k) of the differentiated current, while the second multiplicand is asum of vectors comprising the values l_(k) of the differentiated currentweighted with differences between consecutive voltage values u_(k).

[0023] The equation system (4) can also be solved by means of recursiveparameter estimation methods or Kalman filters. An iterative method may,e.g. for each vector pair m_(n), y_(n), calculate a new approximationΘ_(n) for the parameter vector Θ from the preceding approximationΘ_(n−1), by means of the recursion formula

K=P _(n−1) ·m _(n)·(λ+m_(n) ^(T) ·P _(n−1) ·m _(n))⁻¹

Θ_(n)=Θ_(n−1) +K·(y _(n) −m _(n) ^(T)·Θ_(n−1))

[0024] $\begin{matrix}\begin{matrix}{K = {P_{n - 1} \cdot m_{n} \cdot ( {\lambda + {m_{n}^{T} \cdot P_{n - 1} \cdot m_{n}}} )^{- 1}}} \\{\Theta_{n} = {\Theta_{n - 1} + {K \cdot ( {y_{n} - {m_{n}^{T} \cdot \Theta_{n - 1}}} )}}} \\{P_{n} = {( {E - {K \cdot m_{n}^{T}}} )\frac{P_{n - 1}}{\lambda}}}\end{matrix} & (8)\end{matrix}$

[0025] In this, E is the unit matrix, λ is a weighting factor between0.8 and 0.9 and P_(n) is the so-called precision matrix (start valuee.g. 10³·E or 10⁵·E). K is referred to as a correction factor.

[0026] List of references

[0027]1: line

[0028]2: short circuit point

[0029]3: voltmeter

[0030]4: current differential meter

[0031]5: Rogowski coil

[0032] A: factor

[0033] E: unit matrix

[0034] ƒ_(A): sampling rate

[0035] i(t): time-varying current

[0036] K: correction factor

[0037] L: inductance

[0038] N: number of measurements within the observation window

[0039] R: resistance

[0040] T_(A): time interval between measurements

[0041] u(t): voltage

[0042] u_(k): voltage values

[0043] l(t): time-differentiated current

[0044] l_(k): values of the time-differentiated current

[0045] α₀, α₁: parameter values

[0046] λ: weighting factor

[0047] Θ: parameter vector

1. A method for measuring the resistance R and the inductance L of aline, in which a voltage u across the line and a time derivative i ofthe current through the line are measured at a rate 1/T_(A) in order toobtain a series of N voltage values u₀, u₁, . . . u_(N) and a series ofN values i₀, i₁, . . . i_(N) of the differentiated current,[characterized in that] wherein N≧3 and the voltage values and thevalues of the differentiated current are substituted into an equationsystem with N−1 equations of the form u _(n) =u _(n−1)+α₀ ·i _(n)+α₁ ·i_(n−1), with parameters $\alpha_{0} = {\frac{R}{A} + L}$$\alpha_{1} = {\frac{R}{A} - L}$

where A≅2/T_(A) or${A = {\omega_{0} \cdot {\cot ( {\frac{\omega_{0}}{2} \cdot \frac{T_{A}}{2}} )}}},$

and [in that] wherein the resistance R and the inductance L are foundfrom the equation system.
 2. The method as claimed in claim 1,[characterized in that] wherein N≧3, and [in that] wherein theresistance R and the inductance L are found by adjustment computation.3. The method as claimed in claim 2, [characterized in that] wherein theresistance R and the inductance L are found by solving $\begin{pmatrix}\alpha_{0} \\\alpha_{1}\end{pmatrix} = {\lbrack {\sum\limits_{n = 1}^{N}\quad {m_{n}m_{n}^{T}}} \rbrack^{- 1} \cdot {\sum\limits_{n = 1}^{N}\quad {m_{n}y_{n}}}}$with $m_{n} = \begin{pmatrix}i_{n} \\i_{n - 1}\end{pmatrix}$

and y_(n)=(u_(n)−u_(n−1)).
 4. The method as claimed in [one of claims1-3] claim 1, [characterized in that] wherein the equation system issolved iteratively by calculating, for a plurality of n, anapproximation Θ_(n) for $\Theta = \begin{pmatrix}\alpha_{0} \\\alpha_{1}\end{pmatrix}$

from the recursion formula K=P _(n−1) ·m _(n)·(λ+m _(n) ^(T) ·P _(n−1)·m _(n))⁻¹Θ_(n)=Θ_(n−1) +K·(y_(n) −m _(n) ^(T)·Θ_(n−1))$P_{n} = {( {E - {K \cdot m_{n}^{T}}} )\frac{P_{n - 1}}{\lambda}}$

with $m_{n} = \begin{pmatrix}i_{n} \\i_{n - 1}\end{pmatrix}$

and y_(n)=(u_(n)−u_(n−1)) where E is the unit matrix, λis a weightingfactor, in particular between 0.8 and 0.9, and P_(n)is a matrix with astart value, preferably, between 10³·E and 10⁵·E.
 5. The method asclaimed in [one of the preceding claims] claim 1, [characterized inthat] wherein the first derivative of the current is measured using aRogowski coil.
 6. The method as claimed in [one of the preceding claims]claim 1, [characterized in that] wherein A=2/T_(A.)
 7. The method asclaimed in [one of claims 1-5] claim 1, [characterized in that] wherein${A = {\omega_{0} \cdot {\cot ( {\frac{\omega_{0}}{2} \cdot \frac{T_{A}}{2}} )}}},$

where ω₀ is the angular frequency of a voltage along the line.
 8. Theuse of the method as claimed in [one of the preceding claims] claim 1for measuring the impedance of an electrical power line.